Identifying as yet undetected high-energy sources in the γ -ray sky is one of the declared objectives of the Fermi Large Area Telescope (LAT) Collaboration. We develop a Bayesian mixture model which is capable of disentangling the high-energy extra-galactic sources present in a given sky region from the pervasive background radiation. We achieve this by combining two model components. The first component models the emission activity of the single sources and incorporates the instrument response function of the Fermi γ -ray space telescope. The second component reliably reflects the current knowledge of the physical phenomena which underlie the γ -ray background. The model parameters are estimated using a reversible jump MCMC algorithm, which simultaneously returns the number of detected sources, their locations and relative intensities, and the background component. Our proposal is illustrated using a sample of the Fermi LAT data. In the analysed sky region, our model correctly identifies 116 sources out of the 132 present. The detection rate and the estimated directions and intensities of the identified sources are largely unaffected by the number of detected sources.